關於類晶體的發現，流傳著許多有趣而且值得深度反省的傳說，值得有興趣的讀者自己去探索。準晶體的發現過程中有許多有趣且發人深省的故事，值得有興趣的讀者去探究探索。."2022年春節期間，我偶然看到一本書的書名，讀到一篇文章的標題，《第二種不可能》
說實話，我的英文很差，沒耐心看長篇文章，甚至都沒想過要讀完這本書的英文評論。作為一個英語不太好，但仍然具有批判性思維的人，我不明白這是什麼意思。所以我多次的請教了美國的劉教授。我完全誤解了這本書名字背後的深層意義，而這個美麗的誤解讓我開始回顧我所有關於平舖的藝術作品，並開始深入研究彭羅斯平鋪和準晶體。
到目前為止，已經有數百種具有不同成分和不同冶金熱處理工藝的準晶體被生產出來，而彭羅斯鑲嵌則具有無限的鑲嵌結構,,, 一類是二維數學平鋪問題，幾何 位置 (相對的)固定，內部結構不受限制，具有不隨時間改變的特性 。
一個是至少有兩個(或更多)具有不同特徵的不同原子(原子簇。)的三維物理問題，(含有數百種不同的化學組合),熱熔解液化階段到冷卻階段，在達到穩定的過程中，有不同熱處理的時間變化的問題。
這兩個事件僅通過五重對稱性和變化和組合週期性的特徵,進行只有局部的校正，沒有大局的考量。
我們只能從這兩個事件相同的二維幾何相關性來理解這一點，但它們具有相同的拓撲特徵，即六個五邊形的集合，並且這些集合的所有頂點都是都位於彭羅斯磚中的特定位置 A 形十邊形的中心。
这六个五边形组成的图片隐藏了 3-D 十二面体特征的结构。
Penrose tiling is a mathematical problem in trigonometric drawing, while two-dimensional TEM imaging of quasicrystal surface atom arrays is a physical problem. The only connection between these two problems is that they both possess five-fold rotational symmetry and exhibit non-periodic characteristics, but also possess indeterminate periodicity that defies simple description. How these two problems are connected remains a major question awaiting a logical explanation.
We will begin by analyzing and interpreting the characteristics of regular pentagonal arrays from quasicrystal surface observations (HRTEM) and Penrose tilings, aiming to gain a more precise understanding of their relationship and unresolved issues. 彭羅斯平鋪是三角繪圖中的數學問題，而準晶體表面原子陣列的二維透射電子顯微鏡成像則是物理問題。這兩個問題之間的唯一聯繫在於，它們都具有五重旋轉對稱性，表現出非週期性，但又具有難以簡單描述的不確定週期性。這兩個問題如何關聯，仍然是亟待邏輯解釋的重大問題。
我們將首先分析和解讀準晶體表面觀測（HRTEM）和彭羅斯平鋪中正五邊形陣列的特徵，旨在更準確地理解它們之間的關係以及尚未解決的問題。
2
Future work: We have successfully explained the connection between the 2D Penrose framework and the quasicrystal atomic array in the 2D space. We believe that our proposed 3D quasicrystal model can well illustrate the possibility of the 3D version from two different perspectives. We tried to construct a Penrose-tiled three-dimensional structure using four different three-dimensional hexahedrons with acute angles of 36 and 72 degrees, as shown in Figures A, B, C, and D. One is a semi-three-dimensional tiling formed by stacking two-dimensional tiling models, and the other is to construct a true three-dimensional dodecahedron
未來工作：我們特意製作了網格大小根據黃金比例不同的五邊形和菱形網格，，如圖所示。We called this grids as topological grids for cell comparison , we also tried a similar grid for rhombic dodecahedron , The rhombic dodecahedron is a space-filling polyhedron, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space . The rhombus faces of a rhombic dodecahedron have two different angles: an acute angle and an obtuse angle. The acute angle is approximately 70.53 degrees, and the obtuse angle is approximately 109.47 degrees, according to Wikipedia. The ratio of the lengths of the diagonals of each rhombus is 1:√2, but the grid was not fiting the tiling structure,
我們成功解釋了二維彭羅斯網格框架與二維空間中準晶體原子陣列之間的連結。我們相信，我們提出的三維準晶體模型可以從兩個不同的角度來很好地闡述三維版本的可能性。
我們嘗試用4種不同的銳角分別為36度和72度的三維六面體，構建一個彭羅斯式平舖的三維結構，如圖A、B、C、D所示，一種是由二維平鋪模型堆積而成的半三維平鋪，另一種是構建一個真正的三維十二面體。
我們首先將彭羅斯鑲嵌規範化為二元鑲嵌，並利用五邊形和菱形的拓樸網格形狀 在此標準二元鑲嵌展開，並與準晶原子陣列的拓樸網格絡形狀進行比較。比較结果 的一致性使得我們得出結論：二維空間中的兩個事件（標準Penrose鑲嵌網格和準晶原子陣列）在結構類型上確實彼此密切相關。在此比較结果得出拓樸網格頂點皆位於彭羅斯鑲嵌的十邊形type -a型中心，且皆位於準晶原子陣列中Pd原子或其他型態成核中心上.
The model can now easily explain the bonding of quasicrystals without being restricted to covalent or metallic bonds, simply by utilizing the concept of van der Waals forces to uniformly position atoms on a uniform Penrose tiling framework, or even just uniformly distributed in side an dodecahedron spce..
該模型現在可以輕鬆解釋準晶體的鍵合，而不受共價鍵或金屬鍵的限制，只需利用范德華力的概念，將原子僅僅均勻地分佈在十二面體三維空間中。the topologic matching between the atomic array of quasicrystal and Penrose tiling confirmed that there are quite associated with each other,
原子在2-D彭羅斯鑲嵌框架中均勻分佈的觀測，為彭羅斯式三維十二面體結構的三維(視覺)化提供了可能性。這次嘗試的理論模型和猜想，也試圖將利用一對菱形鑲嵌的二維彭羅斯鑲嵌觀測，拓展到利用一對六面體的三維猜想.
本3-D quasi crystal model 研究中的2D轉3D模型在數學邏輯上過於牽強，僅僅將一個五邊形的二維原子陣列放置在三維十二面體的二維表面上，是一種牽強的猜想。但在現實世界中，我們確實觀察到了十二面體準晶形狀的三維固體，儘管我們並不知道這種十二面體合金內部原子的詳細結構。Now,我們需要做的是將二維表面原子陣列 see作為邊界條件，向內填充十二面體結構的內部，以滿足更精細的二維觀測需求。 The way we are doing is to make different dodecahedron
The observation of that the atoms uniformly located in the Penrose tiling framework provides the possibility of a three-dimensional visible structure of a Penrose style 3-D dodecahedron structure. The theoretical model and conjecture of this attempt also attempts to extend the two-dimensional observation Penrose tiling using a pair of rhombus tiles to a three-dimensional conjecture using a pair of hexahedron. 六面體
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There are many interesting and introspective stories about the discovery of quasi-crystals, which are worth exploring for interested readers. During the 2022 Spring Festival, I came across the title of a book, "The Second Impossibility: To be honest, my English is very poor and I don’t have the patience to read long articles. I haven’t even thought about reading the English reviews of this book. As someone whose English is not very good but still has critical thinking ability, I didn’t understand what this meant. So I asked Professor Liu from the United States. I completely misunderstood the deeper meaning behind the book’s title, and this beautiful misunderstanding led me to review all my artwork on tilings and to delve deeper into Penrose tilings and quasicrystals.
So far, hundreds of quasicrystals with different compositions and different metallurgical heat treatment processes have been produced, while Penrose tessellations have infinite tessellation structures.
關於quasi晶體的發現，流傳著許多有趣而且值得深度反省的傳說，值得有興趣的讀者自己去探索. 準晶體的發現過程中有許多有趣且發人深省的故事，值得有興趣的讀者去探究。."2022年春節期間，我偶然看到一本書的書名，讀到一篇文章的標題，《第二種不可能》
說實話，我的英文很差，沒耐心看長篇文章，甚至都沒想過要讀完這本書的英文評論。作為一個英語不太好，但仍然具有批判性思維的人，我不明白這是什麼意思。所以我多次的請教了美國的劉教授。我完全誤解了這本書名字背後的深層意義，而這個美麗的誤解讓我開始回顧我所有關於平舖的藝術作品，並開始深入研究彭羅斯平鋪和準晶體。
到目前為止，已經有數百種具有不同成分和不同冶金熱處理工藝的準晶體被生產出來，而彭羅斯鑲嵌則具有無限的鑲嵌結構,,, 一類是二維數學平鋪問題，幾何 位置 (相對的)固定，內部結構不受限制，具有不隨時間改變的特性 。
一個是至少有兩個(或更多)具有不同特徵的不同原子(原子簇。)的三維物理問題，(含有數百種不同的化學組合),熱熔解液化階段到冷卻階段，在達到穩定的過程中，有不同熱處理的時間變化的問題。
這兩個事件僅通過五重對稱性和變化和組合週期性的特徵,進行只有局部的校正，沒有大局的考量。. 彭羅斯平鋪是三角幾何繪圖的數學問題，準晶體surface atom array 二維TEM影像是物理問題。這兩個問題之間的唯一聯繫是它們都具有五重旋轉對稱性，並且都具有非週期性特徵,但also have無法簡單形容的不確定週期性。這兩個問題如何的連結在一起,卻是一個等待邏輯性解釋的大問題。
我們將從準晶表面觀測（HRTEM）和彭羅斯平鋪中分析和解釋正五邊形陣列的特徵開始，更準確地理解它們之間的關係,和尚未解決的問題。我們只能從這兩個事件相同的二維幾何相關性來理解這一點，但它們具有相同的拓撲特徵，即六個五邊形的集合，並且這些集合的所有頂點都是都位於彭羅斯磚中的特定位置 A 形十邊形的中心。
这六个五边形组成的图片隐藏了 3-D 十二面体特征的结构。
Penrose tiling is a mathematical problem in trigonometric drawing, while two-dimensional TEM imaging of quasicrystal surface atom arrays is a physical problem. The only connection between these two problems is that they both possess five-fold rotational symmetry and exhibit non-periodic characteristics, but also possess indeterminate periodicity that defies simple description. How these two problems are connected remains a major question awaiting a logical explanation.
We will begin by analyzing and interpreting the characteristics of regular pentagonal arrays from quasicrystal surface observations (HRTEM) and Penrose tilings, aiming to gain a more precise understanding of their relationship and unresolved issues.
We provide a comprehensive solution to quasi crystal related problems, rather than just providing partial answers to individual issues.
彭羅斯平鋪是三角繪圖中的數學問題，而準晶體表面原子陣列的二維透射電子顯微鏡成像則是物理問題。這兩個問題之間的唯一聯繫在於，它們都具有五重旋轉對稱性，表現出非週期性，但又具有難以簡單描述的不確定週期性。 個問題如何關聯，仍然是亟待邏輯解釋的重大問題。
我們將首先分析和解讀準晶體表面觀測（HRTEM）和彭羅斯平鋪中正五邊形陣列的特徵，旨在更準確地理解它們之間的關係以及尚未解決的問題。我們為準晶體問題提供全面的解決方案，而不是僅僅為個別問題提供部分答案。We make a special comparison between these two events based on the principles of quintuple symmetry and the golden ratio.我們根據五重對稱和黃金分割的原則對這兩件事Penrose tiling and atomic array, 進行了特殊的比較。
We propose two contradictory and complementary models, both of which are transformed from 2D models to 3D models: one is based on 2D HRTEM atomic arrays, and the other is based on Penrose tiling. Both converted three-dimensional models are actually empty dodecahedron shell models, except that the 12 faces of the dodecahedron have surface-patterned pentagons, with a five-fold symmetric Penrose tessellation or a five-fold symmetric atomic arrangement pattern.
我們提出了兩個相互矛盾又互相補充的模型，它們都是從二維模型轉換為三維模型：一個基於二維HRTEM原子陣列，另一個基於彭羅斯平鋪。兩個轉換後的三維模型其實都是空的十二面體殼模型，只不過十二面體的12個面上具有表面圖案化的五邊形，具有五重對稱彭羅斯鑲嵌或五重對稱原子排列圖案。
但這兩個模型可以利用表面圖案作為邊界條件來推斷內部結構，從而建構真實的三維模型。
Dear Ms. :Thank you for your invitation. I am sorry that I do not have enough money to support this trip. My daughter is unwilling to pay for my travel expenses to attend the remote conference because of my bad eyesight seeing, or even the registration fee. All my research results will all be posted on face book.
Sincerely,
C. Kung
